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1
JEE Main 2023 (Online) 1st February Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

If $$y(x)=x^{x},x > 0$$, then $$y''(2)-2y'(2)$$ is equal to

A
$$4(\log_{e}2)^{2}+2$$
B
$$8\log_{e}2-2$$
C
$$4\log_{e}2+2$$
D
$$4(\log_{e}2)^{2}-2$$
2
JEE Main 2023 (Online) 1st February Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$f(x) = 2x + {\tan ^{ - 1}}x$$ and $$g(x) = {\log _e}(\sqrt {1 + {x^2}} + x),x \in [0,3]$$. Then

A
there exists $$\widehat x \in [0,3]$$ such that $$f'(\widehat x) < g'(\widehat x)$$
B
there exist $$0 < {x_1} < {x_2} < 3$$ such that $$f(x) < g(x),\forall x \in ({x_1},{x_2})$$
C
$$\min f'(x) = 1 + \max g'(x)$$
D
$$\max f(x) > \max g(x)$$
3
JEE Main 2023 (Online) 31st January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $$y=f(x)=\sin ^{3}\left(\frac{\pi}{3}\left(\cos \left(\frac{\pi}{3 \sqrt{2}}\left(-4 x^{3}+5 x^{2}+1\right)^{\frac{3}{2}}\right)\right)\right)$$. Then, at x = 1,

A
$$2 y^{\prime}+\sqrt{3} \pi^{2} y=0$$
B
$$y^{\prime}+3 \pi^{2} y=0$$
C
$$\sqrt{2} y^{\prime}-3 \pi^{2} y=0$$
D
$$2 y^{\prime}+3 \pi^{2} y=0$$
4
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$

and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$

have a common extreme point, then $a+2 b+7$ is equal to :
A
6
B
$\frac{3}{2}$
C
3
D
4
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